On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow

摘要

Uniform shear flow is a paradigmatic example of a nonequilibrium fluid stateexhibiting non-Newtonian behavior. It is characterized by uniform density andtemperature and a linear velocity profile $U_x(y)=a y$, where $a$ is theconstant shear rate. In the case of a rarefied gas, all the relevant physicalinformation is represented by the one-particle velocity distribution function$f({\bf r},{\bf v})=f({\bf V})$, with ${\bf V}\equiv {\bf v}-{\bf U}({\bf r})$,which satisfies the standard nonlinear integro-differential Boltzmann equation.We have studied this state for a two-dimensional gas of Maxwell molecules withgrazing collisions in which the nonlinear Boltzmann collision operator reducesto a Fokker-Planck operator. We have found analytically that for shear rateslarger than a certain threshold value the velocity distribution functionexhibits an algebraic high-velocity tail of the form $f({\bf V};a)\sim |{\bfV}|^{-4-\sigma(a)}\Phi(\phi; a)$, where $\phi\equiv \tan V_y/V_x$ and theangular distribution function $\Phi(\phi; a)$ is the solution of a modifiedMathieu equation. The enforcement of the periodicity condition $\Phi(\phi;a)=\Phi(\phi+\pi; a)$ allows one to obtain the exponent $\sigma(a)$ as afunction of the shear rate. As a consequence of this power-law decay, all thevelocity moments of a degree equal to or larger than $2+\sigma(a)$ aredivergent. In the high-velocity domain the velocity distribution is highlyanisotropic, with the angular distribution sharply concentrated around apreferred orientation angle which rotates counterclock-wise as the shear rateincreases.